Abstract
Iterative solution method for mesh approximation of an optimal control problem of a system governed by a linear parabolic equation is constructed and investigated. Control functions of the problem are in the right-hand side of the equation and in Neumann boundary condition, observation is in a part of the domain. Constraints on the control functions, state function and its time derivative are imposed. A mesh saddle point problem is constructed and preconditioned Uzawa-type method is applied to its solution. The main advantage of the iterative method is its effective implementation: every iteration step consists of the pointwise projections onto the segments and solving the linear mesh parabolic equations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Dautov, R. Kadyrov, E. Laitinen, A. Lapin, J. Pieskä, and V. Toivonen, “On 3D dynamic control of secondary cooling in continuous casting process,” Lobachevskii J. Math. 13, 3–13 (2003).
M. Gunzburger, E. Ozugurlu, J. Turner, and H. Zhang, “Controlling transport phenomena in the Czochralski crystal growth process,” J. Cryst. Growth 234, 47–62 (2002).
I. Neitzel and F. Troltzsch, “On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints,” in ESAIM: Control, Optimisation and Calculus of Variations. doi 10.1051/cocv:2008038 (2008).
K. Deckelnick and M. Hinze, “Variational discretization of parabolic control problems in the presence of pointwise state constraints,” J. Comp.Math. 29, 1–15 (2011).
A. Lapin and E. Laitinen, “Explicit algorithms to solve a class of state constrained parabolic optimal control problems,” Russian J. Numer. Analysis Math.Modeling 30 (6), 351–362 (2015).
A. Lapin and E. Laitinen, “Iterative solution methods for parabolic optimal control problem with constraints on time derivative of state function,” WSEAS Recent Advances inMathematics: Mathematics and Computers in Science and Engineering 48, 72–74 (2015).
M. Benzi, G. Golub, and J. Liesen, “Numerical solution of saddle point problems,” Acta Numerica 14, 1–137 (2005).
S. Schaible and J-C. Yao, “Recent developments in solution methods for variational inequalities and fixed point problems,” in Proceedings of the American Conference on Applied Mathematics (AMERICANMATH’ 10) (Harvard University, Cambridge, 2010), pp. 142–145.
C. Pan, “On generalized preconditioned Hermitian and skew-Hermitian splitting methods for saddle point problems,” WSEAS Transactions onMathematics 11 (12), 1147–1156 (2012).
M. Fortin and R. Glowinski, Augmented Lagrangian Methods (North-Holland, Amsterdam, 1983).
R. Glowinski and P. LeTallec, Augmented Lagrangian Operator-Splitting Methods in Nonlinear Mechanics (SIAM, Philadelphia, PA, 1989).
N. Pop, “Saddle point formulation of the quasistatic contact problems with friction,” in Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation (Athens, 2007), pp. 250–254.
C. Graser and R. Kornhuber, “Nonsmooth Newton methods for set-valued saddle point problems,” SIAM J. Numer. Anal. 47, 1251–1273 (2009).
A. Lapin, “Preconditioned Uzawa type methods for finite-dimensional constrained saddle point problems,” Lobachevskii J.Math. 31 (4), 309–322 (2010).
E. Laitinen, A. Lapin and S. Lapin, “On the iterative solution of finite-dimensional inclusions with applications to optimal control problems,” Comp.Methods in Appl. Math. 10 (3), 283–301 (2010).
E. Laitinen, A. Lapin, “Iterative solution methods for a class of state constrained optimal control problems,” AppliedMathematics 3 (12), 1862–1867 (2012).
E. Laitinen and A. Lapin, “Iterative solution methods for the large-scale constrained saddle point problems,” Numerical methods for differential equations, optimization, and technological problems, Comp. Meth. Appl. Sci. 27, 19–39 (2013).
A. Lapin, M. Khasanov, “Iterative solution methods for mesh approximation of control and state constrained optimal control problem with observation in a part of the domain,” Lobachevskii J. Math. 35 (3), 241–258 (2014).
J.-L. Lions and E. Magenes, Problemes aux limites non homogenes et applications (Dunod, Paris, 1968).
Ph. G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978).
I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976).
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations (Springer, 1997).
E. Laitinen, A. Lapin, and S. Lapin, “Iterative solution methods for variational inequalities with nonlinear main operator and constraints to gradient of solution,” Lobachevskii J. Math. 33 (4), 364–371 (2012).
Author information
Authors and Affiliations
Corresponding author
Additional information
Submitted by A. M. Elizarov
Rights and permissions
About this article
Cite this article
Lapin, A., Laitinen, E. Preconditioned Uzawa-type method for a state constrained parabolic optimal control problem with boundary control. Lobachevskii J Math 37, 561–569 (2016). https://doi.org/10.1134/S1995080216050085
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080216050085